5 Marks Questions

Question 15 Marks

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?

Answer

By taking LCM of time taken (in minutes) by Sonia and Ravi, We can get the actual number of minutes after which they meet again at the starting point after both start at the same point and at the same time, and go in the same direction.
$18 = 2 \times 3 \times 3 = 2 \times 3 ^ { 2 }$

12 = 2 $\times$ 2 $\times$ $3 = 2^2 $$\times$ 3

LCM $( 18,12 ) = 2 ^ { 2 } \times 3 ^ { 2 } = 36$
Therefore, both Sonia and Ravi will meet again at the starting point after 36 minutes.

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Question 25 Marks

Explain why 7 $\times$ 11 $\times$ 13 + 13 and 7 $\times$ 6 $\times$ 5 $\times$ 4 $\times$ 3 $\times$ 2 $\times$ 1 + 5 are composite numbers.

Answer

So, the factors of 156 are 2 $\times$ 2 $\times$ 3 $\times$ 13

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Question 35 Marks

Prove that $\sqrt 5 $ is irrational.

Answer

Let us prove $\sqrt 5 $ irrational by contradiction.
Let us suppose that $\sqrt 5 $ is rational. It means that we have co-prime integers a and b (b ≠ 0)
Such that $\sqrt 5 = \frac{a}{b}$
$\Rightarrow $ b $\sqrt 5 $=a
Squaring both sides, we get
$\Rightarrow $ $5b ^2 =a^2 ... (1)$
It means that 5 is factor of $a^2$
Hence, 5 is also factor of a by Theorem. ... (2)
If, 5 is factor of a , it means that we can write a = 5c for some integer c .
Substituting value of a in (1) ,
$5b^2 = 25c^2$
$\Rightarrow b^2 =5c^2$
It means that 5 is factor of $b^2$ .
Hence, 5 is also factor of b by Theorem. ... (3)
From (2) and (3) , we can say that 5 is factor of both a and b .
But, a and b are co-prime .
Therefore, our assumption was wrong. $\sqrt 5 $ cannot be rational. Hence, it is irrational.

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